Least-energy Solutions to a Non-autonomous Semilinear Problem with Small Diiusion Coeecient

Least-energy Solutions to a Non-autonomous Semilinear Problem with Small Diffusion Coefficient

Existence and multiplicity of positive solutions for a class of semilinear elliptic system with nonlinear boundary conditions

This study concerns the existence and multiplicity of positive weak solutions for a class of semilinear elliptic systems with nonlinear boundary conditions. Our results is depending on the local minimization method on the Nehari manifold and some variational techniques. Also, by using Mountain Pass Lemma, we establish the existence of at least one solution with positive energy.

A Review of Travelling Wave Solutions

In this paper we review the existence of diierent types of travelling wave solutions u(x; t) = (x ? ct) of degenerate non-linear reaction-diiusion equations of the form u t = D(u)u x ] x + g(u) for diierent density-dependent diiusion coeecients D and kinetic part g. These include the non-linear degenerate generalized Fisher-KPP and the Nagumo equations. Also, we consider an equation whose diius...

Long-time Behavior of Scalar Viscous Shock Fronts in Two Dimensions

We prove nonlinear stability in L 1 of planar shock front solutions to a viscous conservation law in two spatial dimensions, and obtain an expression for the asymptotic form of small perturbations. The leading-order behavior is shown rigorously to be governed by an eeective diiusion coeecient depending on forces transverse to the shock front. The proof is based on a spectral analysis of the lin...

A two-phase free boundary problem for a semilinear elliptic equation

In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary‎. ‎We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose ...